Method and device for monitoring and controlling appliances and installations with a bi-or multifunctional operating range

ABSTRACT

A Method and apparatus for monitoring and controlling appliances and systems having a two-dimensional or multidimensional operating envelope. A map of the operating envelope or of a part of it onto a unit domain or the operating envelope of another appliance is used for monitoring and control. The apparatus produces this map. The method and the apparatus allow monitoring and control which can be carried out substantially in the same way for appliances whose operating envelopes have different shapes.

The invention relates to a method and an apparatus for monitoring andcontrolling an appliance or a system comprising a number of appliances,having a two-dimensional or multidimensional operating envelope, whoselimits represent different technical, procedure, financial orcontractual restrictions.

One example of an appliance having a two-dimensional (planar) operatingenvelope, which is referred to as an operating characteristic, is acompressor. The gas volume flow V and the enthalpy difference H arechosen, for example, as characteristic variables for the operating pointof a compressor. These characteristic variables are used as co-ordinateaxes for representing the operating characteristic, which is defined asthe totality of all those points (operating points) at which thecompressor may be operated in the V′H plane. In practice, the operatingcharacteristic is mainly in the form of a curved polygon. The sides ofthe polygon are referred to as bounds whose infringement results incertain secondary conditions no longer being satisfied. One of thebounds is, for example, the pump bound, which defines a minimum gasvolume flow at which there is still protection against pump surges;beyond the pump bound, operation of the compressor becomes unstable.Further bounds correspond to the maximum rating of the compressor drive,the maximum flow rate, and the minimum and the maximum rotation speed.

The monitoring of a compressor has two tasks:

1. Preventative monitoring, to protect the equipment against dangerousand unstable operating conditions, for example protection against pumpsurging.

2. Functional monitoring in order, for example, to set a specificcompressor rotation speed as a function of the operating state (forexample the load), or to maintain a predetermined nominal value.

Known monitoring systems for compressors are based on the measurement ofa number of characteristic variables for the compressors and theircontrollers. If the operating point infringes what is referred to as acontrol line, which runs parallel to a bound, and at a specific distancefrom it, in the operating characteristic, suitable measures are taken inorder to return the operating point to an area on this side of thecontrol line once again.

In EP-B-0 332 888, the rate of movement of the operating point in thedirection of the pump limit in the operating characteristic isdetermined; if a pump control line, which occurs at a variable distancefrom the pump limit depending on the speed, is infringed, a blow-out orbypass valve is opened quickly, in addition to the normal blow-outcontrol.

U.S. Pat. No. 3,994,623 discloses the linking of a number of originallyindependent control loops. A method is proposed which can be carried outusing a cascade circuit. The method comprises the control loops for therotation speed, the feed pressure and the gas volume flow being linkedto one another, with the output signal from each outer loop representingthe input signal for the next inner loop.

In known control methods, a separate control loop is provided for eachof the control lines. If a control line is infringed, measures aretaken, which depend on the respectively infringed control line and itsspecific shape. These methods are thus based on an independentdescription of each individual control line. If the operating point isin an area close to two bounds (a “corner” of the operating envelope),control of the operating point is very complicated. The control loopsassociated with the various bounds thus often operate independently ofone another and conflict with one another, or else the interaction ofthe various control loops is incomplete.

Similar difficulties also occur when controlling other appliances orsystems which comprise a number of appliances, having an operatingenvelope with two, three or more dimensions.

In the case of a three-dimensional operating envelope, the boundaryconditions, which must not be infringed in either direction, aregenerally represented as, possibly curved, surfaces. The operatingenvelope is then an irregular polyhedron with curved side surfaces. Ifthe operating point is in the vicinity of more than one of the sidesurfaces, this results in a complicated control response. In the generalcase of an N-dimensional operating envelope, the boundary conditions cangenerally be represented as, possibly curved, (N−1)-dimensionalhyperplanes. In this case as well, particular control difficulties arisewhen the operating point approaches more than one of the hyperplanes.

A first object of the present invention is to provide a standard methodfor monitoring or controlling the position of the operating point in atwo-dimensional or multidimensional operating envelope, which isindependent of the specific shape of the operating envelope and whichavoids said difficulties when the operating point approaches the limitsof the operating envelope. The method is intended to allow the user tobe warned reliably of unacceptable or dangerous operating conditions(monitoring), and to avoid them, or to maintain a predetermined nominalvalue (control).

This object is achieved by a method for monitoring or controlling anappliance or a system comprising a number of appliances, having atwo-dimensional or multidimensional operating envelope, in which,according to the invention, the operating envelope, or a part of it, ismapped onto a unit domain or onto the operating envelope of anotherappliance for monitoring or control.

In this method, an operating point or set point in the operatingenvelope is advantageously transformed by means of the map to a point inthe unit domain or in the operating envelope of the other appliance. Themonitoring or control is in this case carried out on the basis of theposition of the image of the operating point which results from the map.

The term unit domain in this case means intrinsically any desired regionwhich is selected and is cohesive as an entity. For two-dimensionaloperating envelopes, for example, the unit circle or a half plane canadvantageously be used as the unit domain. The unit sphere or asemi-infinite body advantageously carries out this role for operatingenvelopes with more dimensions.

If a map is made of the operating envelope onto a unit domain, themethod according to the invention can be used to identify when theoperating point is approaching a limit of the operating envelope, sincethe image of the operating point approaches a limit of the unit domain.One advantage of the method according to the invention is in this casethat, provided the unit domain, the mapping rule and the co-ordinaterepresentation of the image are selected appropriately, the situationwhere the image of the operating point is approaching a limit of theunit domain can be defined and controlled by monitoring a single limit.

A further advantage of the method according to the invention is that themonitoring of the operating point of different appliances can be carriedout in a standard way by using differently shaped operating envelopes.

If the operating envelope of the appliance to be monitored is mappedonto the operating envelope of another appliance, the operating point ofthe appliance to be monitored can be monitored and controlled in thesame way as for the other appliance. In particular, in this way, methodswhich have been proven for monitoring the operating point of a specificappliance can be transferred to the monitoring of the operating point ofany other desired appliances with operating envelopes of any shape inthe same dimension.

The method according to the invention is suitable for a large number offields of application, in particular for monitoring a compressor.

In the method according to the invention, one or more process parametersare advantageously determined by means of the map, or the inversetransformation with respect to it.

In one advantageous refinement, the method for monitoring according tothe invention contains the following steps:

a) determination of the operating point in the operating envelope;

b) transformation of the operating point by means of a stored map rulefor the map;

c) outputting the co-ordinates of the image of the operating point.

In one advantageous refinement, a control method according to theinvention contains the following steps:

a) determination of the operating point in the operating envelope;

b) transformation of the operating point by means of a stored map rulefor the map;

c) comparison of the position of the image of the operating point withbounds, control lines or with the image of a set point in the imagearea;

d) determination of control parameters which define a control action;

e) carrying out the control action.

For a two-dimensional operating envelope, the method according to theinvention can advantageously be implemented such that the map transformsa polygon, which is similar to the operating envelope, to a unit domainor an operating envelope of another appliance. The mapping can thenadvantageously be carried out by means of a function defined by theChristoffel-Schwarz integral. In this case, it is advantageous toapproach the operating envelope from the inside. This means that thepolygon which is similar to the operating envelope is located entirelyin the interior of the operating envelope.

A further object of the invention is to provide an apparatus forcarrying out the method for monitoring and control.

This object is achieved by an apparatus for monitoring or controlling anappliance or a system comprising a number of appliances having atwo-dimensional or multidimensional operating envelope in which,according to the invention, means are provided for determining andstoring parameters which are required to produce a map which maps theoperating envelope, or a part of it, into a unit domain or into theoperating envelope of another appliance.

The apparatus advantageously furthermore contains means for producingthe map for any desired point in the operating envelope.

Furthermore, it is advantageous for the apparatus to contain means forcomparing images of two points located in the operating envelope, withthe comparison being carried out in the unit domain or in the operatingenvelope of the other appliance.

In a further advantageous refinement, the apparatus contains means forcomparing an image of a point which is located in the operating envelopewith at least one line (in particular a control line or bound) in theunit domain or in the operating envelope of the other appliance.

Furthermore, the apparatus may contain means for determining one or moreprocess parameters by means of the map of the operating envelope ontothe unit domain or onto the operating envelope of the other appliance.

An apparatus just for monitoring the operating point of an appliance orof a system comprising a number of appliances in this caseadvantageously contains the following parts:

a) means for determining the operating point of the appliance or of thesystem comprising a number of appliances;

b) means for producing a map of the operating point, which maps theoperating envelope or a part of the operating envelope onto a unitdomain or the operating envelope of another appliance;

c) means for outputting the co-ordinates of the image of the operatingpoint.

An apparatus for controlling an appliance or a system comprising anumber of appliances can advantageously contain the following parts:

a) means for determining the operating point of the appliance or of thesystem comprising a number of appliances;

b) means for producing a map of the operating point, which maps theoperating envelope or a part of the operating envelope onto a unitdomain or the operating envelope of another appliance;

c) means for determining parameters which define a control action;

d) means for carrying out the control action.

Finally, an advantageous refinement of an apparatus for control maycontain the following parts:

a) a first computation model for transformation of a set point to apoint in the unit domain or in the operating envelope of the otherappliance;

b) a measuring unit for determining the position of the operating point;

c) a second computation module for transformation of the operating pointto a point in the unit domain or of the operating envelope of the otherappliance;

d) a comparison unit, which compares the transformed operating pointwith the transformed set point or with at least one bound or controlline of the transformed operating envelope;

e) a monitoring unit for determining the parameters required forcontrol,

f) and an execution unit for carrying out the operations required forcontrol.

In this case, the first and second computation models may also beidentical; in this case, the (single) computation module transforms boththe set point and the operating point.

A method according to the invention for monitoring and controlling theposition of the operating point, and the principles required tounderstand this method, will be explained in more detail in thefollowing text.

From the mathematical point of view, an operating envelope is generallyrepresented by a integral cohesive region. It is then always possible toproduce a continuous map, which can be differentiated and is reciprocal,of the operating envelope onto the interior of a unit domain (that is tosay of another integral cohesive region).

Furthermore, mapping of the operating envelope onto a unit domain, andsubsequent mapping of the unit domain onto the operating envelope ofanother appliance, in principle allows any N-dimensional operatingenvelope to be mapped onto an N-dimensional operating envelope ofanother appliance.

Particularly in the case of two-dimensional operating envelope, theco-ordinate axes may be regarded as the real and imaginary axes in thecomplex number plane. Riemann's mapping rule for complex analysis thenensures that a unique (reciprocal) and conformal map onto the interiorof the unit circle exists for each integral cohesive region.

In order to carry out the monitoring and control method, a mapping rulefor mapping the operating envelope or a part of it onto a unit domain orthe operating envelope of another appliance is first of all stored insome suitable form.

The stored mapping rule may, for example, be a computation rule forcalculating the image of each point within the operating envelope, or apart of it. In the simplest case, this computation rule is an analyticalformula or an approximation formula for the map. Alternatively, themapping rule may, for example, comprise a table which contains theassociated image points for selected points in the operating envelope(original image point), possibly together with an interpolation rule onthe procedure to be adopted for intermediate points in the operatingenvelope.

The stored mapping rule depends on the specific shape of the operatingenvelope, and is characteristic of the appliance to be controlled.Generally, the storage process is carried out only once for a specificappliance, provided the appliance characteristics do not change (forexample due to ageing).

For the actual monitoring procedure, the operating point is first of alldetermined in the operating envelope. This step is expediently carriedout by means of measuring appliances which are suitable for detectingthe characteristic variables of the appliance to be controlled. Themeasured values are converted to co-ordinate values in the operatingenvelope. The operating point then consists of a set of N co-ordinatevalues, corresponding to the N dimensions of the operating envelope. Inthe next step, the operating point is mapped by means of the storedmapping rule. The image of the operating point is now in the form of aset of N values of transformed co-ordinates, which no longer need tocorrespond directly to the characteristic variables of the appliance tobe monitored.

In the next step, comparisons are carried out on the basis of theposition of the image of the operating point. To this end, for example,the position of the image of the operating point is determined withrespect to the boundary of the image area. Alternatively, oradditionally, the position of the image of the operating point can becompared with the position of the image of a set operating point, inwhich case the set operating point may be defined once or may becontinuously matched to the operating requirements. It is likewisepossible for the rate at which the image of the operating point isapproaching the limit of the image area, or the image of the setoperating point, to be determined as well.

Suitable control parameters are defined on the basis of suchinformation. This may be done, for example, by first of all determiningthe direction, in the transformed co-ordinates, in which the image ofthe operating point is intended to be moved within the image area as aresult of the control action. This direction may be, for example, thedirection vector from the image of the operating point to the image ofthe set operating point, or a normal vector, pointing into the interiorof the image area, on the boundary of the image area.

Now, firstly, the direction vector just determined can be transformed bymeans of the map which is in the inverse of the stored mapping rule tothe co-ordinates of the original operating envelope, and a specificcontrol action can be defined from the resultant original image of thedirection vector. In this case, the inverse mapping process can becarried out, in a similar way to the original mapping process, by meansof a computation rule or a table from original image and image points orfrom original image and image vectors. If the inverse mapping process iscarried out by means of a table, then this table may be the same as thatwhich was used for the original mapping of the operating point.

On the other hand, the determined direction vector in transformedco-ordinates can instead of this be used directly for defining theparameters for the specific control action. It is advantageous for thispurpose for a table to have previously been defined in which themovement direction of the image of the operating point to be expected inthe transformed co-ordinates has been entered for specific controlactions, which are to be carried out once. Suitable control actions forrecording in such a table are, in particular, those actions during whichonly a single operating parameter is changed. A specific control action,which may comprise the changing of a number of operating parameters, canthen be defined by means of this table.

Finally, in the last step of the proposed method, the control actiondefined in this way is carried out on the basis of the determinedcontrol parameters.

Furthermore, an apparatus for monitoring the operating point accordingto the invention will now be described in more detail.

First of all, this apparatus contains means for determining theoperating point. These means may be, for example, measuring applianceswhich, at their output, produce a signal which represents a uniquemeasure of the value of a characteristic variable. The signals producedby the measuring appliances thus represent the operating point.

Furthermore, the apparatus contains means for producing the map of theoperating point. These means may, for example, comprise ananalogue/digital converter, a computation unit and a memory unit. Theanalogue/digital converters convert the signals which represent theoperating point to a corresponding digital value. The operating point isnow represented by N digital values.

A computation unit uses these values and a mapping rule which is storedin a memory unit to produce new digital values, which represent theco-ordinates of the image of the operating point in the unit domain. Themapping rule may in this case be located in the memory unit, for examplein the form of a computation rule, or may be in the form of a table withan interpolation rule. Instead of being carried out digitally, thecalculation can also be carried out by analogue means, without usinganalogue/digital converters. The means for producing the map of theoperating point then contain an analogue computation device.

Finally, means are provided for outputting the co-ordinates of theoperating point. These may be, for example, a screen for producing agraphic display of one or more co-ordinates of the image of theoperating point, or digital/analogue converters with downstream displayinstruments.

An apparatus which is intended to be used not just for monitoring anddisplaying the operating point, but also for controlling it, containsfurther components.

Thus, furthermore, an apparatus such as this has means for determiningthe parameters which define a control action. These may comprise, forexample, a computation unit, a memory unit and a digital/analogueconverter. The computation unit uses the position of the image of theoperating point in transformed co-ordinates to calculate the parameterswhich define a control action; in the process, it accesses computationrules stored in the memory unit, or tables for calculating theseparameters. The digital/analogue converter converts these parametersinto signal values. The means for carrying out the control action thenuse these signal values to directly influence the response of theappliance or of the system to be controlled.

Exemplary embodiments of the invention will be explained with referenceto the attached drawings, in which:

FIG. 1 shows an outline sketch of an apparatus for monitoring, forpreventing the boundary of the characteristic from being infringed andfor set value control of an appliance or of a system according to thepresent invention.

FIG. 2 shows the operating characteristic of a compressor,

FIG. 3 shows the unit circle as a unit domain,

FIG. 4 shows an outline sketch of the monitoring of a compressor,

FIG. 5 shows the upper half-plane as a unit domain,

FIG. 6 shows a rectangular operating characteristic, and

FIG. 7 shows a polygonal operating characteristic.

EXAMPLE 1

In this example, FIG. 1 will be used to explain an apparatus accordingto the invention for monitoring and for preventing infringement of theboundary of the characteristic of an appliance or of a system.

The controlled appliance 1000 in FIG. 1 is equipped with a unit 1090 forpredetermining the operating envelope, with a unit 1100 for defining theforward and reverse transformation of the operating envelope and fordetermining the forward and reverse transformation parameters, and witha unit 1110 for storing the transformation rules, which are required ina unit 1120 for transformation of the operating envelope, with all itsbounds, to a unit domain.

A measuring unit 1010 determines the position of the operating point(actual point). A functional unit 1020 transforms the operating point toa point in the unit domain. This unit 1020, if necessary, alsodetermines parameters such as the rate and direction of movement of theoperating point.

A comparison unit 1030 compares the transformed operating point with thebound or control line of the transformed operating envelope.

A further unit 1040 determines the control variables and the newposition of the control lines in the transformed area. This unitdetermines the parameters required to carry out the control process, andthe new position of the control line, for dynamic control systems as afunction of the rate and direction of movement of the operating point.

The reverse transformation of the parameters for the control actions tobe carried out takes place in a unit 1050. An execution unit 1060 in thecontrol system carries out the control actions on the control members1070.

The entire control process can be followed using an output andvisualization unit 1080.

EXAMPLE 2

In this example, FIG. 1 will be used to explain a general description ofan apparatus for set value control and for monitoring an appliance or asystem, using the method according to the invention.

The units 1000 to 1120 in FIG. 1 are described in Example 1. The setvalue S is predetermined in an input unit 1130, and is transformed in acomputation unit 1140 to a point in the unit domain.

The transformed set value is compared in the comparison unit 1030 withthe transformed actual value, and with the control lines in the unitdomain.

Before this comparison, the intervals between the actual value and theset value, the actual value and the control lines, and the set value andthe control lines can be added or multiplied using appropriate penaltyfunctions (D. Himmelblau, Applied Nonlinear Programming, McGraw-HillBook Co., 1972, Part III, Chapter 7; N. Staroselsky and L. Ladin, Moreeffective control for centrifugal gas compressors operating in parallel,Paper 86-GT-204, Am. Soc. Mec. Eng. 1986, page 7 and FIG. 8). The objectof these functions is to weight the intervals depending on whether theactual value is relatively close to or well away from the control lines.

This provides an interaction between the set value control and theprevention of the infringement of the boundary of the characteristic asin Example 1.

EXAMPLE 3

This example describes an apparatus and a method according to theinvention for checking, monitoring and controlling a compressor with atwo-dimensional operating envelope.

FIG. 2 shows, schematically, a typical two-dimensional operatingenvelope for a compressor. This operating envelope is also referred toas the operating characteristic 2. The co-ordinates in thisrepresentation are the gas volume flow V′ in m³/h and the enthalpydifference H in J. The pressure ratio r=P_(d)/P_(i) often used insteadof the enthalpy difference, with P_(i) being the inlet pressure, andP_(d) being the outlet pressure.

The operating characteristic 1 is bounded by the sides s₁ to s₄, whichare referred to as the bounds. The side s_(i) is the pump limit, beyondwhich the operation of the compressor becomes unstable. The side s₂comprises the lowest values from two operating lines—the maximum drivepower line and the maximum rotation speed line for the compressor. Theside s₃ corresponds to the maximum possible feed rate, and the side s₄,corresponds to the operating line for the minimum compressor rotationspeed. The comers of the characteristic, i.e. the intersections of thesides, are annotated b₁ to b₄. Furthermore, FIG. 2 shows the controllines r₁ to r₄, the operating point B and the set operating point S.

In conventional methods for compressor monitoring and control, theposition of the operating point B with respect to one or more of thecontrol lines is monitored separately for each control line.

FIG. 3 shows, as the unit domain, the unit circle 3 with the radiusR_(E)=1. A map, which maps the characteristic 1 from FIG. 2 onto theunit circle 3 in FIG. 3, is annotated ψ₁; it is assumed that this isknown. The images of the corners b₁ to b₄ are annotated a₁ to a₄(a_(i)=ψ₁(b_(i)), i=1, . . . , 4). The images of the bounds s₁ to s₄ arecorrespondingly annotated s₁′ to s₄′, and are in the form of circulararc sections. The image of the operating point under the map ψ₁ isannotated B′=ψ₁(B), and the image of the set operating point iscorrespondingly annotated S′.

The procedure for monitoring and controlling the compressor using themethod according to the invention will be explained with reference toFIG. 4. FIG. 4 shows, schematically, an apparatus for controlling acompressor 4000. The functions of the units 4000 to 4140 correspond tothose of the units 1000 to 1140 in FIG. 1.

The measuring unit 4010 comprises a number of measuring instruments forthe inlet pressure, the operating pressure, the gas volume flow etc.,e.g. 4011, 4012 and 4013, and analogue-digital and digital-digitalconverters, for example 4014, 4015 and 4016.

The transformation rules are stored in the memory unit 4110 in the formof coefficients, computation processes etc., and a table, whichassociates the selected points in the operating envelope 2, shown inFIG. 2, with their image points in the unit circle 3 in FIG. 3. Theoperating point B is now monitored and controlled as follows:

The instantaneous values of the gas volume flow V′ and of the pressureratio r are determined using the measuring instruments mentioned above.The values of V′ and r define the operating point B, and they aresupplied to the computation unit 4020. The transformation rules storedin the memory unit 4110 and a table of original image and image pointsare used by the computation unit 4020 to calculate, the position of theimage B′ of the operating point in the unit circle 4. The co-ordinatesof B′ are, for example, in the form of polar co-ordinates R_(B), φ_(B)at the output of the computation unit 4020.

In a similar way, the unit 4140 calculates the position of the image S′of the set point S, which is predetermined in 4130. Co-ordinates of S′are, for example, polar co-ordinates R_(S), φ_(S).

Furthermore, the co-ordinates of the operating point B′ and of the setpoint S′ in the image area are supplied to the control unit 4030. Thecontrol unit compares the co-ordinates of the image of the operatingpoint B′(R_(B), φ_(B)) with the co-ordinates of the image of thepreviously defined set operating point S′ (R_(S), φ_(S), and with thecontrol line 301 shown in FIG. 3, that is to say with its radius R_(R).

The unit 4040 uses the result of this comparison to calculate adirection vector, along which the operating point will be moved.

In this case, the value R_(R)-R_(B) directly indicates the distancebetween the image of the operating point and the control line in theunit circle, and, in a corresponding manner, 1-R_(B), indicates thedistance from the unit circle boundary. If R_(B) is greater than apreviously defined value R_(w), then an audible warning signal can beemitted, which warns the operator of the compressor 4000 that theoperating point is approaching a boundary of the operating envelope, ina dangerous manner.

Control variables for preventing the boundaries of the compressorcharacteristic from being infringed and for approaching the set valueare determined in the unit 4040, analogously to the examples 1, 2.

If the actual point is in the vicinity of the control line or isapproaching the control line too quickly, then the bypass valve openingposition can be set, for example, in proportion to |R_(R)-R_(B)|.

If the actual point is in the vicinity of the set point, then the changein the compressor rotation speed can be set, for example, in proportionto |B′-S′|, in order to reach the set value.

In the situation where a rapid change in the actual value is intendedand the distance between the operating value and the set value is large,and the compressor drive control is slow to react, the bypass value canalso briefly be actuated.

The direction vector determined in 4040 is now mapped by the inversetransformation unit 4050 in a direction vector in the originalcharacteristic 2. The inverse transformation unit 4050 for this purposeaccesses the table of original image and image points stored in thememory unit 4110. Finally, the control unit 4040 calculates the desiredposition of the bypass valve 4071 and of the compressor rotation speed4072 on the basis of the direction vector obtained in this way, in imageand original image co-ordinates. The bypass valve 4071 and thecompressor drive control 4072 are actuated by the calculated values bymeans of the execution device 4060. This actuation represents the actualcontrol action.

The described compressor control method has at least two advantages overconventional methods. The first advantage is that the monitoring of thepermissible operating conditions is carried out on the basis of a singlebound, namely the unit circle line, and comprises monitoring of a singleparameter, namely R_(B). In a corresponding way, the control process forprotection against unacceptable operating conditions is carried out onthe basis of a single control line, namely the control line 301. Theimage of the characteristic, namely that of the unit circle, in thiscase has no comers. The known problems of controlling the operatingpoint in the vicinity of comers of the characteristic therefore does notoccur when using the described method.

The second advantage is that the calculation steps which are carried outin the control units 4030, 4040, 4050 are independent of the shape ofthe original operating characteristic. These calculation steps can thusbe applied in this same way to a large number of appliances havingdifferent characteristics. Other steps than those described here are, ofcourse, also feasible in this case. Thus, for example, the speed of theimage of the operating point in transformed co-ordinates can also beused for control purposes.

EXAMPLE 4

In this example, the method from Example 3 is modified such that theupper half plane is used as the unit domain rather than the unit circle.FIG. 5 shows the upper half plane 5. This is bounded at the bottom bythe co-ordinate axis, which is annotated x. Once again, it is assumedthat a map ψ₂ is known, which maps the characteristic 1 from FIG. 1 intothe upper half plane 5. The images of the corners b₁ to b₄ are annotateda₁′ to a₄′ (a₁′=ψ₂(b_(i)),i=1, . . . , 4) and the images of the boundss₁ to s₄ are annotated s₁″ to s₄″. Furthermore, FIG. 5 shows ahorizontally running control line 501, which intersects the co-ordinateaxis, which is annotated y, at the point y_(R). The compressor is onceagain monitored and controlled by means of an apparatus as shown in FIG.4. The memory unit 4110 now contains a table, which associates thepoints in the operating characteristic 2 shown in FIG. 2 with imagepoints in the upper half plane 5, and possibly a reciprocal table.

The monitoring and control are carried out in the same way as in Example1, with the following differences: the computation unit 4020 calculatesthe position of the image B″ of the operating point in the upper halfplane. The co-ordinates of the image of the operating point areproduced, for example, as rectangular co-ordinates x_(B), y_(B), at theoutput of the computation unit. The output unit indicates the valuesx_(B) and y_(B); in this case, y_(B) directly indicates the distancebetween the image of the operating point and the boundary of the upperhalf plane. If y_(B) is less than a value y_(W) defined in advance, thenan audible warning signal is emitted. The comparison unit 4030 issupplied with the co-ordinates x_(B) and y_(B). The control process isnow carried out on the basis of these co-ordinates, the co-ordinatesx_(S) and y_(S) of the image of the set operating point, and the controlline 501, in a similar manner as in Example 3.

EXAMPLE 5

In this example, an explicit rule for mapping of the characteristic ontothe upper half plane is specified for an appliance having a rectangularcharacteristic (I. N. Bronstein and K. A. Semendjajew, Taschenbuch derMathematik [Mathematic Manual] 25 Edition, Teubner, Stuttgart 1991).FIG. 6 shows a rectangular characteristic 6. The co-ordinate axes shownare annotated Re z and Im z, and are the real and imaginary axes in thecomplex number plane. The characteristic 6 has the corners$\frac{v_{1}}{2},{- \frac{v_{1}}{2}},{\frac{v_{1}}{2} + {iv}_{2}},{{- \frac{v_{1}}{2}} + {iv}_{2}}$

in this number plane. One such rectangle 6 can be obtained from anygiven rectangular characteristic by rotation and shifting, for exampleby multiplication by a complex number whose magnitude is 1, and additionof a complex number.

One function which maps the complex upper ζ half plane onto therectangle 6, as shown in FIG. 6, in the z-plane, is the function${z = {c\quad {\int_{0}^{\zeta}\quad \frac{t}{\sqrt{\left( {1 - t^{2}} \right)\quad \left( {1 - {\kappa^{2}\quad t^{2}}} \right)}}}}},$

where c is a real positive constant which depends on the size of therectangle, and κ is a real number between 0 and 1, which depends on theratio of the side lengths v₂/v₁ of the rectangle. (A. Hurwitz and R.Courant, Funktionentheorie, [Function theory] 4th Edition,Springer-Verlag, Berlin, pages 437 et sec). The variables c and κ can becalculated numerically for a given rectangle.

The stated function is the reciprocal function of the mapping, accordingto the method, of the operating envelope into the unit domain. The tableof original image and image points quoted in Example 3 above can in thisway be produced easily. The points z in the rectangular operatingcharacteristic are determined numerically, using the quoted formula, fora selection of points ζ=x+iy in the upper half plane. The points z arestored as original image points in the table, and the points ζ arestored as image points.

EXAMPLE 6

In this example, Example 5 is generalized to form a general polygonalcharacteristic. FIG. 7 shows such a characteristic 7, e.g. having sixcorners b₁ to b₆. The corners of a general n-sided polygon are annotatedb₁ to b_(n), and the angles are annotated πg₁, to πg_(n). Theco-ordinate axes are annotated Re z and Im z, and are the real andimaginary axes of the complex number plane as in the previous Example 5.One function, which maps the complex upper half plane onto a polygonwhose angles are πg₁ to πg_(n), is the Christoffel-Schwarz integralz = c₁  ∫₀^(ζ)(t − a₁)^(g₁ − 1)  (t − a₂)^(g₂ − 1)  …  (t − a_(n))^(g_(n) − 1)  t + c₂

where c₁ and c₂ are complex constants (Hurwitz and Courant, et al, pages431 et sec). The real points a₁ to a_(n) are in this case those pointswhich are mapped by this function onto the cornerss of the polygon b₁ tob_(n).

The points a₁ to a_(n) can now be determined using numerical methods forpredetermined corners points b₁ to b_(n).

The stated function is the reciprocal function of the mapping accordingto the method of a polygonal operating characteristic onto the upperhalf plane as the unit domain. A table of original image and imagepoints can be produced using this, in a similar way to that in Example5.

EXAMPLE 7

An example of a method for monitoring the operating point of anappliance or of a system comprising a number of appliances having atwo-dimensional or multidimensional operating envelope is described inthe following text, in which the operating point is monitored in termsof the position of the image of the operating point resulting from amapping process which maps only a part of the operating envelope onto aunit domain.

Once again, a two-dimensional operating characteristic 2 of any givenshape is considered, as shown in FIG. 2. Such a characteristic can beapproximated with sufficient accuracy by a polygon with straight edges.In this case, it is advantageous to choose the polygon such that it isentirely located inside the characteristic.

The monitoring of the operating point and the control of an appliancehaving this characteristic of any given shape can then be carried out onthe basis of the position of the image of the operating point resultingin a mapping process which maps the polygon located in the interior ofthe operating envelope onto the upper half plane. The control process iscarried out such that the operating point is within the polygon in alloperating conditions. This also ensures that the operating point is inthe interior of the actual operating characteristic in all operatingconditions.

The advantage of using a polygon to approximate the operatingcharacteristic is that functions which map the upper half plane and apolygon onto one another can be specified in a general analytical form,as in Example 6.

EXAMPLE 8

A method for monitoring an appliance having a two-dimensional operatingenvelope will be explained in this example, in which the monitoring iscarried out on the basis of the position of the image of the operatingpoint which results from a mapping process which maps the operatingenvelope of the appliance to be controlled onto the operating envelopeof another appliance.

In this case, it is assumed that the operating envelope of the applianceto be controlled has, for example, the shape shown in FIG. 2. It is alsoassumed that another appliance exists, whose operating envelope has, forexample, the shape shown in FIG. 6, and that an efficient, safe andreliable control method is known for this appliance. Finally, atransformation t₁, which maps the operating characteristic 2 from FIG. 2onto a unit domain, and a transformation t₂, which maps the operatingcharacteristic 6 from FIG. 6 onto the same unit domain, are assumed tobe known.

A map of the characteristic 2 onto the characteristic 6 can then becomposed of the maps t₁ and t₂: the desired map is t₂₁=t₂ ⁻¹t₁.

The method for monitoring and controlling the appliance using theoperating characteristic 2 can now be carried out as follows: first ofall, the operating point of this appliance is determined, and theoperating point is mapped into the characteristic 6. The methods (whichare assumed to be known) for controlling the appliance by means of thecharacteristic 6 are now used to determine parameters which define acontrol action for the appliance with the characteristic 2 to becontrolled. This can be done, for example, in the manner described aboveby inverse transformation of a direction vector, obtained on the basisof these methods, in the characteristic 6 in FIG. 6.

One advantage of this method is that the efficient, safe and reliablecontrol method assumed for the appliance having the characteristic 6 canbe applied equally well to an appliance having any given characteristic,such as the characteristic 2. In this way, a control method which isknown per se for a specific appliance having a planar characteristic canbe used to control any given appliance having a planar characteristic.

We claim:
 1. A method for controlling at least one appliance having afirst operating range that is multidimensional, said method comprisingsteps of: mapping at least a part of the first operating range onto asecond operating range of another appliance; and transforming a firstoperating point in the first operating range by a map to a secondoperating point in the second operating range of the another appliancefor control.
 2. The method as claimed in claim 1, wherein at least oneprocess parameter is determined by the map.
 3. The method as claimed inclaim 1, further comprising: determining the first operating point inthe first operating range; transforming the first operating point by astored map rule for the map; and outputting co-ordinates of the secondoperating point.
 4. The method as claimed in claim 1, furthercomprising: determining the first operating point in the first operatingrange; transforming the first operating point by a stored map rule forthe map; determining parameters which define a control action, based onthe co-ordinates of the second operating point; and carrying out thecontrol action.
 5. The method as claimed in claim 1, wherein the firstoperating range is two-dimensional, and wherein the mapping step maps apolygon, which is similar to the first operating range, onto the secondoperating range of the another appliance.
 6. The method as claimed inclaim 1, wherein at least one process parameter is determined by aninverse transformation with respect to the map.
 7. An apparatus forcontrolling at least one appliance, each of said at least one appliancehaving a first operating range that is multidimensional, said apparatuscomprising: means for determining and storing parameters required toproduce a map which maps at least part of the first operating range andan operating point in the first operating range into a second operatingrange of another appliance; means for producing the map; means foroutputting coordinates of an image of the operating point; and means forcarrying out a control action.
 8. The apparatus as claimed in claim 7,further comprising means for producing the map for any point in thefirst operating range.
 9. The apparatus as claimed in claim 7, furthercomprising means for comparing images of two points located in the firstoperating range, with the comparison being carried out in the secondoperating range of the another appliance.
 10. The apparatus as claimedin claim 7, further comprising means for comparing an image of a pointlocated in the first operating range with at least one line in thesecond operating range of the another appliance.
 11. The apparatus asclaimed in claim 7, comprising means for determining at least oneprocess parameter by the map of the first operating range onto thesecond operating range of the another appliance.
 12. The apparatus asclaimed in claim 7, further comprising: a first computation module fortransformation of a set point to a transformed set point in the secondoperating range of the another appliance; a measuring unit fordetermining a position of the operating point; a second computationmodule for transformation of the operating point to a transformedoperating point in the second operating range of the another appliance;a comparison unit for comparison of the transformed operating point withthe transformed set point or with at least one control line of thesecond operating range; a monitoring unit for determining parametersrequired for control; and an execution unit for carrying out operationsrequired for control.
 13. A method for controlling at least oneappliance having an operating range that is multidimensional, saidmethod comprising steps of: mapping at least a part of the operatingrange onto a unit domain; and transforming a first operating point inthe operating range by a map to a second operating point in the unitdomain for control.
 14. The method as claimed in claim 13, wherein atleast one process parameter is determined by the map.
 15. The method asclaimed in claim 13, further comprising: determining the first operatingpoint in the operating range; transforming the first operating point bya stored map rule for the map; and outputting co-ordinates of the secondoperating point.
 16. The method as claimed in claim 13, furthercomprising: determining the first operating point in the operatingrange; transforming the first operating point by a stored map rule forthe map; determining parameters which define a control action, based onthe co-ordinates of the second operating point; and carrying out thecontrol action.
 17. The method as claimed in claim 13, wherein theoperating range is two-dimensional, and wherein the mapping step maps apolygon, which is similar to the operating range, onto the unit domain.18. The method as claimed in claim 13, wherein at least one processparameter is determined by an inverse transformation with respect to themap.
 19. An apparatus for controlling at least one appliance, each ofsaid at least one appliance having a multidimensional operating range,said apparatus comprising: means for determining and storing parametersrequired to produce a map which maps at least part of the operatingrange and an operating point in the operating range into a unit domain;means for producing the map; means for outputting coordinates of animage of the operating point; and means for carrying out a controlaction.
 20. The apparatus as claimed in claim 19, further comprisingmeans for producing the map for any point in the operating range. 21.The apparatus as claimed in claim 19, further comprising means forcomparing images of two points located in the operating range, with thecomparison being carried out in the unit domain.
 22. The apparatus asclaimed in claim 19, further comprising means for comparing an image ofa point located in the operating range with at least one line in theunit domain.
 23. The apparatus as claimed in claim 19, furthercomprising means for determining at least one process parameter by themap of the operating range onto the unit domain.
 24. The apparatus asclaimed in claim 19, further comprising: a first computation module fortransformation of a set point to a transformed set point in the unitdomain; a measuring unit for determining a position of the operatingpoint; a second computation module for transformation of the operatingpoint to a transformed operating point in the unit domain; a comparisonunit for comparison of the transformed operating point with thetransformed set point or with at least one control line of the unitdomain; a monitoring unit for determining parameters required forcontrol; and an execution unit for carrying out operations required forcontrol.